Civil Engineering Reference
In-Depth Information
2
1/2
3
⎛ ⎞
⎡
∞
⎤
ρ
BV
2
ˆ
ˆ
()
()
()
2
()
x
x
∫
H
J
d
σ
=
φ
⋅
⋅
⎜ ⎟ ⎢
⋅
ω
⋅
ω
ω
⎥
(6.21)
yr
yr
⎜ ⎟
⎢
y
y
2
mB
ω
⎥
y
⎝ ⎠ ⎣
y
⎦
0
where
ˆ
()
2
J
J
/
∫
dx
ω
=
φ
(6.22)
y
y
y
L
The non-dimensional frequency response function is given in Eq. 4.25. Neglecting any
aerodynamic mass effects and introducing the notation given in Eqs. 5.24 and 5.25, it is
then given by
1
−
2
⎡
⎤
⎛ ⎞
ω
(
)
ω
ˆ
()
⎢
⎥
H
1
2
i
ω
=−
κ
−
⎜
⎜ ⎟
+
ζ
−
ζ
⋅
(6.23)
y
ae
y
ae
⎢
y
ω
y
ω
⎥
⎝ ⎠
y
y
⎣
⎦
where
2
⎡
⎤
B
ρ
2*
2
P
∫
dx
ω
φ
⎢
⎥
y
4
y
2
⎢
⎥
L
⎡
⎤
⎢
exp
K
⎥
ae
y
2
∫
dx
1
2
1
4
⎢
⎥
2
2
φ
m x
∫
⎡ ⎤
ωφ
⎢
⎥
*
y
P
yy y
κ
2
⎡ ⎤ ⎢
⎥
ω
M
2
⎢ ⎥
4
B
m
ae
ρ
⎢
⎥
L
y
yy
L
exp
⎢ ⎥
⎢
=
⎥
=
=
⋅
⋅
(6.24)
⎢ ⎥
⎢
⎥
2
ζ
2
∫
dx
⎢ ⎥ ⎢
C
⎥
B
φ
ρ
⎢ ⎥
*
ae
y
⎣ ⎦
⎢
⎢
*
2
⎥
y
P
y
ae
y
∫
ω
P
φ
dx
⎥
⎢
⎣ ⎦
1
y
1
y
⎢
2
⎥
L
⎢
2
ω
M
⎥
L
exp
2
⎣
yy
⎦ ⎢
⎥
⎢
⎥
2
m x
∫
ωφ
yy y
L
⎢
⎥
⎣
⎦
Similarly, the standard deviation of the dynamic response in the
z
direction and in
torsion are given by
1/2
2
3
⎡
∞
⎤
BV
⎛ ⎞
ρ
2
ˆ
ˆ
()
()
()
2
()
x
x
∫
H
J
d
σ
=
φ
⋅
⋅
⋅
ω
⋅
ω
ω
(6.25)
⎢
⎥
⎜ ⎟
⎢
zr
zr
z
z
2
mB
ω
⎥
⎝ ⎠
⎣
⎦
z
z
0
1/2
2
4
⎡
∞
⎤
BV
⎛ ⎞
ρ
2
ˆ
ˆ
()
()
()
2
()
x
x
∫
H
J
d
σ
=
φ
⋅
⋅
⋅
ω
⋅
ω
ω
(6.26)
⎢
⎥
⎜ ⎟
⎢
r
r
θ
θ
θ
θ
2
mB
ω
⎥
⎝ ⎠
⎣
⎦
θ
θ
0
where:
